Reciprocal of a Linear Function

The reciprocal of a linear function has the form f(x) = 1/kx-c.

Let use this function as example for better understanding:

f(x) = 1/ x-4

Domain

To find the domain of the function, we have to know the restriction of the x first.

It's easy to find the restriction of x. All you have to do is let the denominator equal to zero.

x-4=0

x=4

So x cannot equal to 4. If it is equal to 4, the function will be undefined.

So the domain is { x € R; x ≠ 4}.

End Behaviour

As x approaches 4 from the right, f(x) approaches positive infinity.

As x approaches 4 from the left, f(x) approaches negative infinity.

The graph approaches the vertical line x=4 but does not cross it. So the curve is discontinuous at this line. This line is called vertical asymptote.

As x approaches positive infinity, f(x) approaches zero from the above as all the values of f(x) are positive.

As x approaches negative infinity, f(x) approaches zero from below as all the values of f(x) are negative.

The graph approaches a horizontal line at x-axis but does not cross it. It has the horizontal asymptote of y=0.

Vertical Asymptote

The graph will not cross the line x=4.

Thus, an equation for the vertical asymptote is x=4.

Horizontal Asymptote

The graph will not cross the line y=0.

Thus, an equation for the horizontal asymptote is y=0.

Range

Same as domain, we have to find the restriction of y first.

The graph of the function shows that f(x) gets close to the line y=0 but never actually touches the line. Therefore the only restriction on the range of f(x) is y not equal to 0.

Range is {y € R; y≠0}

For further details, you can watch these video :)